The History of House Targaryen in Westeros can be traced back to about 300 years before the time of Daenerys Targaryen. It may not sound like a massive history because the Targaryens are not originally from Westeros. They come from this ancient land known as Valyria, which does not exist anymore.
The Targaryens’ civilisation was never completely at peace. They were always faced by rebellions and usurpers. But they had no problems in defeating these uprising up and until Robert Baratheon’s rebellion about 282 years after Aegon’s Conquest. The series of events that led to this rebellion are quite fascinating.
The circumstances of Rhaegar kidnapping Lyanna are not clear. Targaryen loyalists say that they were both acting out of love while Robert Baratheon maintained that Rhaegar violated Lyanna till her death. It is unknown what the truth is and when Ned Stark arrives at the tower where Lyanna is kept in order to rescue her, he finds her almost dead in a pool of blood. Right before she dies in her brother’s arms she makes him promise her something, though the contents of this promise are unknown.Robert Rebellion was won by the rebels at the battle of the trident where Robert killed Rhaegar with a massive strike of his warhammer. The Lannisters who remained neutral all this while sacked King’s Landing and the mad king Aerys Targaryen was killed by his own kingsguard Ser Jaime Lannister. It is believed that the Mountain killed Rhaegar’s wife and children. King Aerys’ wife was taken back to dragon stone by Targaryen loyalists along with her second son Viserys Targaryen. And not so long later she died while giving birth to Daenerys Targaryen who was born during a massive storm and hence has the name Daenerys Stormborn.
Read here for the history of House Stark and Lannister.